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Mersenne numbers, two small results on

This entry presents two simple results on Mersenne numbers11In this entry, the Mersenne numbers are indexed by the primes., namely that any two Mersenne numbers are relatively prime and that any prime dividing a Mersenne number MpsubscriptMpM_{p} is greater than ppp. We prove something slightly stronger for both these results:

Theorem.

If qqq is a prime such that q∣Mpfragmentsqnormal-∣subscriptMpq\!\mid\!M_{p}, then p∣(q-1)fragmentspnormal-∣fragmentsnormal-(q1normal-)p\!\mid\!(q-1).

Proof.

By definition of qqq, we have 2p≡1(modq)superscript2pannotated1pmodq2^{p}\equiv 1\;\;(\mathop{{\rm mod}}q). Since ppp is prime, this implies that 222 has orderppp in the multiplicative group∖⁡ℤq, {0}subscriptℤq0\mathbb{Z}_{q}\mathbin{\setminus}\{0\} and, by Lagrange’s Theorem, it divides the order of this group, which is q-1q1q-1.
∎

Theorem.

If mmm and nnn are relatively prime positive integers, then 2m-1superscript2m12^{m}-1 and 2n-1superscript2n12^{n}-1 are also relatively prime.

Proof.

Let d:=gcd⁡(2n-1,2m-1)assigndsuperscript2n1superscript2m1d:=\gcd(2^{n}-1,2^{m}-1). Since ddd is odd, 222 is a unit in ℤdsubscriptℤd\mathbb{Z}_{d} and, since 2n≡1(modd)superscript2nannotated1pmodd2^{n}\equiv 1\;\;(\mathop{{\rm mod}}d) and 2m≡1(modd)superscript2mannotated1pmodd2^{m}\equiv 1\;\;(\mathop{{\rm mod}}d), the order of 222 divides both mmm and nnn: it is 111. Thus 2≡1(modd)2annotated1pmodd2\equiv 1\;\;(\mathop{{\rm mod}}d) and d=1d1d=1.
∎

Note that these two facts can be easily converted into proofs of the infinity of primes: indeed, the first one constructs a prime bigger than any prime ppp and the second easily implies that, if there were finitely many primes, every MpsubscriptMpM_{p} (since there would be as many Mersenne numbers as primes) is a prime power, which is clearly false (consider M11=23⋅89subscriptM11normal-⋅2389M_{{11}}=23\cdot 89).